Elementary cyclic normal group theory

[eileen6a]

Homework Statement

If G is a finite group and let H be a normal subgroup of G with finite index m=[G]. Show that $a^m\in H$ for all a$\in$ G.

Homework Equations

order of a group equal the order of element.

The Attempt at a Solution

no idea.

 

[jbunniii]

Consider the canonical map $\phi : G \rightarrow G/H$ defined by $g \mapsto gH$. What can you say about $\phi(a^m)$?

P.S. "order of a group equal the order of element" is false unless the group is cyclic and the element is a generator of the group.